in the Calculation of Errors. 101 



Accordingly the modulus of ? is inversely proportional to x; 

 and the weight of — is directly proportional to x 2 . The best 



value of p 12 is Sxy-r-^x 2 ; and the error of this determina- 

 tion has for modulus 



\/$x%l-p*) + Sx 2 = Vl-P 2 - .VS^= \/l-p 2 H- VR 

 (the modulus of x being unity by hypothesis) 



= \/2x Vl^p 2 ^ \/w 

 (n being the number of observations utilized) . 



The laborious multiplication which the formula S^y-r-S^ 2 

 involves may be abridged by grouping the a's in small fasci- 

 cules *. 



(2) More convenient methods of utilizing the data are 

 (i.) that which I have recommended in a former paper f : 

 dividing the sum of the assigned deviations Sa? into the sum 

 of the associated deviations S?/; and (ii.) the method to which 

 Mr. Gralton's statistics lend themselves J : arranging the ob- 

 servations in small groups, taking the quotient Sy-f-Sa? for 

 each group, and the arithmetic mean of all these quotients as 

 the value of p. 



(i.) The modulus of the error incident to the expression 

 Sy-r-Sa? (x assigned and y observed to be associated) is 



\/n(l-/D 2 )-rS#= s/v sjT^P-r- Vn 

 (the modulus for the deviation x being unity) ; worse than 

 the best method in the degree Vir : s/2, or 1*25 times. 



This result may be improved by omitting some of the data; 

 in virtue of the following general theorem : — 



When, instead of the proper iceights of a set of observations 

 Pi 2 ? P2 2 j fyc., another set of weights, q 2 2 , q 2 2 , are used as more 

 convenient, it is in general advantageous to reject some of the 

 given observations ; the last observation admitted, say x m , being 

 determined by the equation 



This equation is derived from the condition that we should 

 stop taking in new observations as soon as the modulus of the 



* The principles which should regulate this sort of approximation may- 

 be gathered from the present writer's paper on the " Determination of 

 the Modulus " in the Philosophical Magazine, 1886, vol. xxi. p. 500. 



t Phil. Mag. Aug. 1892. 



X Proc. Eoy. Soc. vol. xlv. p. 139. 



